Now we go to our general solution, C 1, e to the r x, et cetera.

But here, again, we recognize that we only have the first

two terms in this equation will appear in the solution

because it’s a second order differential equation.

So the solution, then, is C1e to the R1x plus C2e to the R2x.

And substituting these values in,

we get this equation, which I can write like this.

So this is our general solution.

Next, however,

we can evaluate those constants by applying the boundary conditions.

So the first boundary condition i'll apply is this one,

dy by dx evaluated at zero is equal to one.

So, I differentiate this expression with respect to

x to get minus two c two e to the minus 2x And

then I substitute x=0 to evaluate that at 0.

And I get that the value is equal to one.

So therefore, solving that equation,

the constant C2 is equal to minus one half.

Next, I apply the first boundary condition, that y(0) = 0.

So substituting x=0 into this equation,

I get Y of zero is equal to C1 plus C2, from which it follows And

evaluating that at 0, I have y(0) = 0,

from which it follows that C1 is equal to -C2,

but C2, we already know, is -1/2.

Therefore, C1 = 1/2.

So putting those values into the general solution here,

we find that our solution is one-half minus one-half e

to the minus 2x, which I can write a little more neatly as

1 minus e to the minus 2x over 2, and so the answer is b.